Conservation laws and evolution schemes in geodesic, hydrodynamic and magnetohydrodynamic flows

TL;DR

This paper extends the Hamiltonian framework for fluid and magnetofluid flows, revealing new conservation laws and geometric structures, and demonstrates their relevance for numerical simulations in general relativity.

Contribution

It shows that ideal magnetohydrodynamics can be incorporated into Hamiltonian variational principles, leading to new conservation laws and geometric insights for fluid flows.

Findings

Magnetofluids can be described using Hamiltonian methods.

Conservation laws like Kelvin and Alfvén are special cases of Hamiltonian invariants.

Extended Kelvin's theorem to baroclinic and magnetized fluids.

Abstract

Carter and Lichnerowicz have established that barotropic fluid flows are conformally geodesic and obey Hamilton's principle. This variational approach can accommodate neutral, or charged and poorly conducting, fluids. We show that, unlike what has been previously thought, this approach can also accommodate perfectly conducting magnetofluids, via the Bekenstein-Oron description of ideal magnetohydrodynamics. When Noether symmetries associated with Killing vectors or tensors are present in geodesic flows, they lead to constants of motion polynomial in the momenta. We generalize these concepts to hydrodynamic flows. Moreover, the Hamiltonian descriptions of ideal magnetohydrodynamics allow one to cast the evolution equations into a hyperbolic form useful for evolving rotating or binary compact objects with magnetic fields in numerical general relativity. Conserved circulation laws, such as those of Kelvin, Alfv\'en and Bekenstein-Oron, emerge simply as special cases of the Poincar\'e-Cartan integral invariant of Hamiltonian systems. We use this approach to obtain an extension of Kelvin's theorem to baroclinic (non-isentropic) fluids, based on a temperature-dependent time parameter. We further extend this result to perfectly or poorly conducting baroclinic magnetoflows. Finally, in the barotropic case, such magnetoflows are shown to also be geodesic, albeit in a Finsler (rather than Riemann) space.